B. Transcendental Gap as Categorical Incompleteness
We apply the CAS-6 functor \(\mathcal{F}: \mathbf{Var} \to \mathbf{LayeredVect}\) to \(X = K3 \times K3\).
Interaction Level \(L(X)\): \(L(X) = H^4(X, \mathbb{Q})\), dimension 486, focusing on \(p=2\).
Interaction Configuration \(C(X)\): \(C(X)_2 = H^{2,2}(X) \cap H^4(X, \mathbb{Q})\), dimension approximately 404 (rational Hodge classes, accounting for transcendental contributions).
Interaction Weights \(W(X)\): \(W(X)_2 = \im(\cl_2: CH^2(X) \otimes \mathbb{Q} \to H^4(X, \mathbb{Q}))\), dimension 400 from divisor products.
Interaction Probabilities \(P(X)\): Compute
  \[
  P(X)_2 = \frac{\dim W(X)_2}{\dim (H^{2,2}(X) \cap H^4(X, \mathbb{Q}))} \approx \frac{400}{404} \approx 0.9901.
  \]
The gap (404 vs. 400) indicates categorical incompleteness: the cycle class map is not surjective, reflecting transcendental obstructions.
Interaction Stability \(S(X)\): Stability is assessed via deformation invariance in the moduli space of K3 pairs. The transcendental lattice contributes a 4-dimensional subspace that persists under generic deformations, so \(S(X)_2 \approx 400\), as algebraic cycles are stable but miss transcendental classes.
